0,3

a(n) is the number of permutations on [n] that avoid the consecutive pattern 132 (pattern entries must occur consecutively in the permutation).

Asymptotically, a(n)/n! ~ c/r^n where r = 1.2755477364172... is the unique positive root of Integrate[exp(-t^2/2),{t,0,r}] = 1 and c = exp(r^2/2)/r = 1.7685063678958....

Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Advances in Applied Mathematics 30 (2003) 110-125.

Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns.

In the Mathematica code below, a[n, k] is the number of such permutations with first entry k and they are counted recursively by the length, say ell, of the longest increasing left factor L. (For ell >=2 the first entry following L must be < the penultimate entry of L or else a consecutive 132 would occur.) The first sum counts ell=1, the second ell=2, the third ell>=3; m is the penultimate entry of L and j is the first entry in the (reduced) subpermutation following L. Note that j is indexed from 0 to cover the case when L is the entire permutation.

E.g.f. Sum_{n>=0}a(n)x^n/n! = 1/( 1-(Pi/2)^(1/2)*Erf[z/2^(1/2)] )

The first 3 entries of 2431 form a consecutive 132 pattern.

The 4!-a(4) = 8 permutations on [4] that DO contain a consecutive 132 pattern are 1243, 1324, 1423, 1432, 2143, 2431, 3142, 4132. Also, for example, 1342 contains a scattered 1-3-2 pattern but not a consecutive 132.

Clear[a]; a[0, 0] = a[0] = 1; a[n_, 0]/; n>=1 := 0; a[n_, k_]/; k>n := 0; a[n_, k_]/; 1<=k<=n<=2 := 1; a[n_, k_]/; n>=3 := a[n, k] = Sum[a[n-1, j], {j, k-1}] + (n-k)Sum[a[n-2, j], {j, k-1}] + Sum[(n-m) Binomial[m-k-1, ell-3]a[n-ell, j], {ell, 3, n-k+1}, {m, k+ell-2, n-1}, {j, 0, m-ell+1}]; a[n_]/; n>=1 := a[n] = Sum[a[n, k], {k, n}]; Table[a[n], {n, 0, 15}]

(* or, faster *) ExpGfToList[f_, n_, x_] := CoefficientList[Normal[Series[f, {x, 0, n}]] /. x^(pwr_) -> e!*x^pwr, x]; ExpgfToList[1/( 1-(Pi/2)^(1/2)*Erf[z/2^(1/2)] ), 25, z]

Cf. A049774.

Sequence in context: A105072 A022494 A136127 this_sequence A079566 A059685 A000112

Adjacent sequences: A111001 A111002 A111003 this_sequence A111005 A111006 A111007

nonn

David Callan (callan(AT)stat.wisc.edu), Oct 01 2005